M294F09WS_DynSys

# M294F09WS_DynSys - The General Solution to d~x dt = A~x t...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: The General Solution to d~x dt = A~x ( t ) is ~x ( t ) = c 1 e Î» 1 t ~v 1 + c 2 e Î» 2 t ~v 2 where â€¢ Î» 1 and Î» 2 are the eigenvalues of A â€¢ ~v 1 and ~v 2 are the associated eigenvectors â€¢ If an initial condition is specified, ~x (0) = c 1 ~v 1 + c 2 ~v 2 . So c 1 and c 2 are just the coordinates of ~x (0) with respect to the eigenbasis ~v 1 and ~v 2 . Note : We do not consider the case when Î» 1 = Î» 2 (even if A is still diagonalizable) If the eigenvalues are complex: ( Î» = p Â± qi with eigenvectors ~v Â± i~w ) The above equation gives all complex solutions to the system. Say p + iq is an eigenvalue with associated eigenvector ~v + i~w . All real solutions of the system are given by ~x ( t ) = e pt ~w ~v cos( qt )- sin( qt ) sin( qt ) cos( qt ) ~w ~v- 1 ~x (0) Sketching the Phase Portrait Real Eigenvalues: (see diagram on page 407 in the textbook) 1. Draw the x 1- x 2 plane. 2. Draw the straight line which is the eigenspace for each eigenvalue. If the eigenvalue is negative, trajectories tend toward the origin and if the eigenvalue is positive, they...
View Full Document

## This note was uploaded on 12/25/2009 for the course MATH 1920 taught by Professor Pantano during the Spring '06 term at Cornell.

### Page1 / 2

M294F09WS_DynSys - The General Solution to d~x dt = A~x t...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online